1. General Relativity

2. Gravitational and inertial mass are equivalent.
General Relativity
Principle of equivalence:
There is no experiment that will discern the difference between the effect of gravity and the effect of acceleration.
Or…
Gravitational and inertial mass are equivalent.

3. Principle of equivalence:
On Earth:
In space: a = 9.8 m/s/s

4. Principle of equivalence:
You feel Zero “g”s in free fall

5. Apparent Curvature of light:
Accelerating up so fast the lady’s a goner
Not accelerating

6. Apparent Curvature of light:
The sun is too bright normally
In 1919, Sir Arthur Eddington
Eclipse
Light was bent twice as much as Newton’s theory predicted, supporting General Relativity

7. Gravitational Lensing:

8. Gravitational Lensing:

9. Curvature of Space: Now that you understand that gravity bends light…
Understand that it does not.
Light travels in a straight line.
The space itself near a massive object is curved.
Light is the absolute.
It travels at the speed of light.
It travels in a straight line.
Do not adjust your television set…
Re-adjust your brain.

10. Curvature of Space:
Mass distorts space
Analogy for dimensions…

11. Curvature of Space: Geometry is Non-Euclidian
Were the sphere large enough…
Riemann and Einstein… (Science itself)

12. Black Holes:
Light cannot escape…

13. Black Holes: Gravitational Potential per unit mass: V = -GM so PE = Vm
At escape velocity, kinetic = potential
1/2mv2 = GMm substituting c for v: r
r = 2GM where r is the Schwarzschild radius c2

14. Black Holes: Black Holes become so by getting smaller
As r gets smaller, v gets bigger, when v = c it is a black hole
Were the Earth 0.35” in radius it would be a black hole
The sun would be 1.9 miles in radius.
The sun and the earth will never become black holes.
Not all by themselves…

15. Put this in your notes: What is the maximum radius of a black hole that is 30. million times the mass of the sun?
Msun = 1.99 x 1030 kg
r = 2*6.67E-11*30E6*1.99E30/3E82 = x1010 m = 8.8x1010 m
(More than half an AU!)
R = 2*6.67E-11*1.44*1.99E30/3E82 = m
8.8x1010 m

16. What is the mass of a black hole the size of the earth?
r = 6.38 x 106 m
M = rc2/(2G) = 6.38E6*3E82/(2*6.67E-11) = 4.3E33 kg
M = rc2/(2G) = 6.38E6*3E82/(2*6.67E-11) = 4.3E33 kgc
4.3E33 kg

17. Clocks and gravitation:
General relativity predicts that clock A will run faster than clock B…
From Feynman Lectures in Physics

18. Clocks and gravitation:
Ship accelerating up
Observer at bottom of ship
Clocks emit pulses of light
Pulse 1 goes distance L1
Pulse 2 goes distance L2
L2 is shorter than L1
Observer sees ticks closer together in time.
If it always appears to be running faster, it is…
Principle of equivalence says gravity must also cause this.
From Feynman Lectures in Physics

19. This -> Clocks and gravitation:
Principle of equivalence says gravity must also cause this.
This ->
From Feynman Lectures in Physics

20. This -> Clocks and gravitation:
Principle of equivalence says gravity must also cause this.
Is the same as
This ->
g = 9.8 m/s/s
From Feynman Lectures in Physics

21. Clocks and gravitation:
Gravity affects the rate clocks run
High clocks run faster
Low clocks run slower
The twin paradox
Flying in a circle paradox
Red shifted radiation from Quasars

22. Approximate formula for small changes of height:
Clocks and gravitation:
Approximate formula for small changes of height:
Δf = gΔh
f c2
Δf - change in frequency
f - original frequency
g - gravitational field strength
Δh - change in height
c - speed of light

23. What frequency do they tune to at the top?
Put this in your notes: A radio station at the bottom of a 320 m tall building broadcasts at 93.4 MHz. What is the change in frequency from bottom to top?
What frequency do they tune to at the top?
93.4*9.8*320/3E82 = 3.3E-12 MHz
Since low clocks run slow, you would tune to a lower frequency at the top. Basically the same frequency.
3.3E-6 Hz

24. What frequency do they tune to at the top?
A radio station at the bottom of a 320 m tall building near a black hole where g = 2.5 x 1013 m/s/s broadcasts at 93.4 MHz. What is the change in frequency from bottom to top?
What frequency do they tune to at the top?
93.4*2.5E13*320/3E82 = 8.30 MHz
Since low clocks run slow, you would tune to a lower frequency at the top. So you would tune to 93.4 – 8.3 = 85.1 MHz
8.3x106 Hz, 85.1 MHz

25. Δf/f = gΔh/c2, g = Δfc2/fΔh 8.5 x 1012 m/s/s
Two trombonists, one at the top of a 215 m tall tower, and one at the bottom play what they think is the same note. The one at the bottom plays a Hz frequency, and hears a beat frequency of 5.2 Hz. What is the gravitational field strength?? For us to hear the note in tune, should the top player slide out, or in? (Are they sharp or flat)
Δf/f = gΔh/c2, g = Δfc2/fΔh
8.5 x 1012 m/s/s
8.5 x 1012 m/s/s, out, sharp

26. Gravitational Time Dilation
Δt - Dilated time interval
Δto - Original time interval
Rs - Schwarzschild radius
r - Distance that the clock is from the black hole

27. Put this in your notes: A graduate student is 5
Put this in your notes: A graduate student is 5.5 km beyond the event horizon of a black hole with a Schwarzschild radius of 9.5 km. If they are waving (in their frame of reference) every 3.2 seconds, how often do we see them waving if we are far away?
t = 3.2/√( 1-9.5/15) ≈ 5.3 s
5.3 s

28. A graduate student is in orbit 32.5 km from the center of a black hole. If they have a beacon that flashes every 5.00 seconds, and we (from very far away) see it flashing every 17.2 seconds, what is the Schwarzschild radius of the black hole?
17.2 = 5.00/√( 1-Rs/32.5)
Rs = 32.5(1-(5.00 s)2/(17.2 s)2) = 29.8 km
29.8 km

29. A graduate student is in orbit 316 km from the center of a black hole with a Schwarzschild radius of 186 km.
We (from very far away) see their beacon flashing every 7.8 seconds. How fast do they see it flashing?
7.8 = to/√( 1-186/316)
to = 7.8√(1-186/316) = 5.0 s
5.0 s